Question: Find the largest integral value of $x$ which solves: $\frac{1}{3}<{\frac{x}{5}}<{\frac{5}{8}}$
Solution: Multiplying by $5$, we have $\frac53<x<\frac{25}8$. Writing this with mixed numbers, we have $1\frac23 < x < 3\frac18$, so the possible integer values for $x$ are $2$ and $3$. Of these, the larger value is $\boxed{3}$.

(Note that only the second inequality matters here. We deal with the first inequality merely to show that there actually are integers satisfying both inequalities!)